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EQUATION OF STATE OF PLASMA OF METALS IN
NANOPARTICLES PRODUCTION USING LASER ABLATION
R.V. Davydov1
1
Saint Petersburg State Polytechnic University, St. Petersburg, Russia
ANNOTATION
To analyze the physical processes at high energy densities, when laser is used, an adequate
description the thermodynamic property of matter over a broad region of states including the
normal conditions and plasma at high pressures and temperatures is required. For describing
the thermodynamic properties of metals in nanoparticles production using laser ablation a
semi-empirical equation of state model is proposed. To verify this model, an equation of state
of aluminum was constructed. Using this equation was calculated shock adiabat for aluminum
and compared with experimental data. The equation of state constructed for aluminum by
proposed model adequately describes the experimental data over the entire range of pressures
generated in shock waves. The model of laser ablation of aluminum in water is in a good
match with experiment.
INTRODUCTION
There is a growing interest in the nanofabrication of materials and their applications in
various fields of life and technology, such as electronics, energy generation, health care and
storage. A great deal of progress in this field has relied on the use of lasers. Production of
nanoparticles can be done in several ways; one of them is laser ablation [1]. Evidence shows
that this method is superior to other methods. Indeed, laser ablation in liquids, which consists
of the pulverization of a solid target in liquid environment, gives a unique opportunity to
solve the toxicity problems. In contrast to chemical nanofabrication methods, laser ablation
can be performed in a clean, well-controlled environment, such as deionized water, giving rise
to the production of ultrapure nanomaterial. The use of these particles decreases toxicity risks,
which is especially important in vivo bio sensing and imaging applications [2].
To analyze the physical processes at high energy densities, when laser is used, an adequate
description of the thermodynamic properties of matter over a broad region of states including
the normal conditions and plasma at high pressures and temperatures is required. For that a
semi-empirical equation of state model is proposed.
MODEL OF EQUATION
The thermodynamic properties of metals are described by the Helmholtz free energy,
which is considered as a sum of three components, describing the thermal contributions of
electrons ( ) and atoms ( ) and the elastic part of the interaction at T = 0 K ( ):
The first component is calculated within the framework of the Thomas–Fermi model with
quantum, exchange and oscillation corrections [3].The advantage of this model is easy
expansion on mixtures and materials in the liquid medium, but it needed some corrections at
normal conditions and very high pressures. For that, two other components of the equation of
state are calculated by empirical way.
The second component is given by formula:
Here
is the value of Gruneisen coefficient under normal conditions and coefficients A,
B, C and D are determined comparing with available thermodynamic data for metals on high
pressures.
The third component is given by interpolation formula [4]:
The value of coefficient
and
are determined by expressions:
Here
is the specific volume at P = 0 and T = 0 K,
Bohr atomic radius,
is the atomic mass unit (amu),
is the Hartree energy,
is the
is the atomic mass (in amu),
is the
atomic number of an element.
To determine the coefficients
we must solve the problem of minimization of the root-
mean-square deviation of pressure at points
from the results of
calculation by the Thomas–Fermi model with corrections subject to the conditions for the
pressure, bulk modulus and its first and second derivative with respect to pressure at
:
The problem of conditional minimization is solved with the introduction of Lagrange
factors [5]. The values of the parameters
,
,
,
are fitted by iterations so as to satisfy
under normal conditions the tabular value of specific volume
and the values of isentropic
compression modulus and its pressure derivative determined by the data of dynamic
measurements.
RESULTS
To verify this model, an equation of state of aluminum was constructed. Using this
equation was calculated shock adiabat for aluminum and compared with experimental data.
The coefficients of the equation of state that optimally generalize the available
thermodynamic information for aluminum are as follows:
The comparison showed that the equation of state constructed for aluminum by proposed
model adequately describes the experimental data over the entire range of pressures generated
in shock waves.
For pressures less than 103 GPa the maximum error of the model was about 5%, for
pressures between 103 and 104 – 15%. For greater pressures, the errors were noticeable,
because Thomas–Fermi model, which was used for describing the thermal contributions of
electrons, did not take into account shell effects in atoms. That is why for receiving correct
results for extremely high pressure we must use more complicated models, which are take in
account shell effects, for example modified Hartree-Fock-Slater model and others. That will
be considering in next papers.
However, using this model of equation of state for modeling laser ablation of aluminum in
water by method in study [6] allowed to obtain very similar and adequate results. This facts
show that proposed model can be used for modeling laser ablation of other metals.
REFERENCES
1. G.N. Makarov, Phys. Usp. 56 (2013), p. 643-682.
2. P. Mulvaney, T. Linnert, A.J. Henglein, Phys. Chem. 95 (1991), p. 7843-7846.
3. A.F. Nikiforov, V.G. Novikov, V.B. Uvarov, Quantum-Statistical Models of Hot Dense
Matter and Methods for Computation Opacity and Equation of State; Progress in
Mathematical Physics 37, Birkhauser Verlag: Basel, 2005, p. 285-294.
4. K.V. Khishchenko, Tech. Phys. Lett. 30 (2004), p. 829-831.
5. A.V. Bushman, V.E. Fortov, I.V. Lomonosov, In High Pressure Equations of State: Theory
and Applications, North-Holland: Amsterdam, 1991, p. 249.
6. Y. Cao, X. Zhao, Y.C. Shin, Analysis of nanosecond laser ablation of aluminum with and
without phase explosion in air and water, Journal of Laser Applications 25 (2013), p. 35-42.